Experimental Mathematics

The subgroups of {$M\sb {24}$}, or how to compute the table of marks of a finite group

Götz Pfeiffer

Abstract

Let G be a finite group. The table of marks of G arises from a characterization of the permutation representations of G by certain numbers of fixed points. It provides a compact description of the subgroup lattice of G and enables explicit calculations in the Burnside ring of G. In this article we introduce a method for constructing the table of marks of G from tables of marks of proper subgroups of G. An implementation of this method is available in the GAP language. These computer programs are used to construct the table of marks of the sporadic simple Mathieu group $M_{24}$. The final section describes how to derive information about the structure of G from its table of marks via the investigation of certain Möbius functions and the idempotents of the Burnside ring of G. Tables with detailed information about $M_{24}$ and other groups are included.

Article information

Source
Experiment. Math. Volume 6, Issue 3 (1997), 247-270.

Dates
First available in Project Euclid: 17 March 2003

Permanent link to this document
http://projecteuclid.org/euclid.em/1047920424

Mathematical Reviews number (MathSciNet)
MR1481593

Zentralblatt MATH identifier
0895.20017

Subjects
Primary: 20D30: Series and lattices of subgroups
Secondary: 20D08: Simple groups: sporadic groups

Keywords
Burnside ring table of marks subgroup lattice Mathieu groups

Citation

Pfeiffer, Götz. The subgroups of {$M\sb {24}$}, or how to compute the table of marks of a finite group. Experiment. Math. 6 (1997), no. 3, 247--270. http://projecteuclid.org/euclid.em/1047920424.


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